Gravitational dynamics: A novel shift in the Hamiltonian paradigm

TL;DR

This paper reinterprets gravitational dynamics within the Hamiltonian framework as a gauge covariant Lie derivative, offering new insights into classical and quantum gravity and potential links to gauge theory concepts.

Contribution

It introduces a novel gauge covariant Lie derivative interpretation of gravitational evolution, connecting Hamiltonian constraints to geometrical transformations on phase space.

Findings

Simplifies calculations in classical general relativity.

Provides a new perspective for quantum gravity quantization.

Suggests potential extensions of gauge theory concepts to gravity.

Abstract

It is well known that Einstein's equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that {\em time} evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in {\em spatial} directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomorphism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the `integrable' (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the `double copy' ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.